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Chapter 2: Problem 69

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{1}{2 x}$$

Short Answer

Expert verified
The function is odd.

Step by step solution

01

Understand the Definitions

A function is **even** if for all numbers \(x\) in the domain of \(f\), \(f(-x) = f(x)\). It is **odd** if \(f(-x) = -f(x)\). Our task is to determine which of these properties applies to the function \(f(x) = \frac{1}{2x}\). If neither holds, then the function is neither even nor odd.
02

Calculate \(f(-x)\)

We start by substituting \(-x\) into the function. \[ f(-x) = \frac{1}{2(-x)} = \frac{-1}{2x} \]
03

Compare \(f(-x)\) with \(f(x)\)

Now compare \(f(-x) = \frac{-1}{2x}\) to \(f(x) = \frac{1}{2x}\).It follows that:- \(f(-x) eq f(x)\), so the function is not even.- \(f(-x) = -f(x)\) since \(\frac{-1}{2x} = -\frac{1}{2x}\), so the function is odd.
04

Conclusion: Determine Function Type

Since \(f(-x) = -f(x)\) is true, the function \(f(x) = \frac{1}{2x}\) is an odd function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Function Evaluation
Function evaluation involves substituting a specific input value into a function to find the output. In our exercise, we need to evaluate the function at \(-x\) to check if it's even or odd.
We began with the function \(f(x) = \frac{1}{2x}\). To evaluate \(f(-x)\), substitute \(-x\) into the function in place of \(x\):
  • Original function: \(f(x) = \frac{1}{2x}\)
  • Evaluated function: \(f(-x) = \frac{1}{2(-x)} = \frac{-1}{2x}\)
With this substitution, you can see how the output changes when the input is \(-x\) instead of \(x\). This evaluation is crucial because it allows us to compare outputs to determine properties like whether the function is even or odd.
Exploring Function Properties
Functions have various properties, including symmetry. Symmetry helps identify if a function is even or odd.
Understanding whether a function is even or odd gives insight into its behavior:
  • A function is **even** if \(f(-x) = f(x)\) for all \(x\) in the domain. Graphically, it means the function is symmetrical about the y-axis.
  • A function is **odd** if \(f(-x) = -f(x)\). This gives the graph rotational symmetry of 180 degrees about the origin.
  • If neither condition is met, the function is neither even nor odd.
For \(f(x) = \frac{1}{2x}\), we evaluated \(f(-x) = \frac{-1}{2x}\) and compared it to \(f(x)\):
  • \(f(-x) eq f(x)\), so \(f(x)\) is not even.
  • \(f(-x) = -f(x)\), confirming that \(f(x)\) is odd.
These comparisons of \(f(x)\) and \(f(-x)\) highlight the function鈥檚 property as being odd.
Relating Precalculus Concepts to Even and Odd Functions
In precalculus, examining even and odd functions helps students understand function behavior deeply.
These concepts form a foundation for more complex mathematical ideas:
  • Even functions simplify calculus work as they often have predictable derivatives and integrals.
  • Odd functions provide interesting characteristics to their calculus when analyzing areas under curves or rotating around axes.
  • The familiarity with even and odd functions aids in exploring polynomial, trigonometric, and other advanced functions later in calculus.
The function \(f(x) = \frac{1}{2x}\) demonstrates an odd function, revealing how such functions transform in a symmetric fashion around the origin. Recognizing these properties in precalculus builds confidence, preparing students for the intricacies of calculus.

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Most popular questions from this chapter

If \(f(x)=k, k \neq 0\) is a constant function and \(g(x)=m x+b, m \neq 0\) is a linear function, then determine the range of the composition \((f \circ g)(x)\)

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=\frac{1}{2} x^{2}+4 x$$

Complete the following. (a) Write an absolute value inequality involving \(f(x)\) that satisfies the given restriction. (b) Solve the absolute value inequality for \(x\). \(f(x)=5 x-12\) must be less than 0.1 unit from \(10 .\)

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{1}{x+1}, g(x)=3-6 x$$

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=x^{5}-2 x^{3}$$
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